A note on the asymptotics for the randomly stopped weighted sums

Yang Yang,Xing Fang Fang Huang,Xi Xi Xi Shi

doi:10.15388/na.2018.2.4

Yang Yang, Xing Fang Fang Huang + Show 1 more

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https://doi.org/10.15388/na.2018.2.4

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Abstract

Let {Xi , i ⩾ 1} be a sequence of identically distributed real-valued random variables with common distribution FX; let {θi , i ⩾ 1} be a sequence of identically distributed, nonnegative and nondegenerate at zero random variables; and let τ be a positive integer-valued counting random variable. Assume that {Xi , i ⩾ 1}, {θi , i ⩾ 1} and τ are mutually independent. In the presence of heavy-tailed Xi's, this paper investigates the asymptotic tail behavior for the maximum of randomly weighted sums Mτ = max1 ⩽ k ⩽ τ ∑ki = 1θi Xi under the condition that {θi , i ⩾ 1} satisfy a general dependence structure.

Highlights

Let {Xi, i 1} be a sequence of identically distributed real-valued random variables (r.v.s) with common distribution FX ; let {θi, i 1} be a sequence of nonnegative and nondegenerate at zero r.v.s, which may be arbitrarily dependent; and let τ be a positive integer-valued r.v
Under the conditions that are independent and identically distributed (i.i.d.) and {θi, i 1} are arbitrarily dependent, some earlier works have been devoted to the investigation of the asymptotic tail behavior for the finite or infinite number of randomly weighted sums or their maximum, i.e., P(Sn > x) and P(Mn > x) (or P(S∞ > x) and P(M∞ > x)) as x → ∞
We are interested in the asymptotic behavior for the tail probability of a random number of randomly weighted sums and their maximum defined in (1), which is motivated by two recent papers [15] and [4]

Summary

Introduction

Let {Xi, i 1} be a sequence of identically distributed real-valued random variables (r.v.s) with common distribution FX ; let {θi, i 1} be a sequence of nonnegative and nondegenerate at zero r.v.s, which may be arbitrarily dependent; and let τ be a positive integer-valued r.v. Are independent and identically distributed (i.i.d.) and {θi, i 1} are arbitrarily dependent, some earlier works have been devoted to the investigation of the asymptotic tail behavior for the finite or infinite number of randomly weighted sums or their maximum, i.e., P(Sn > x) and P(Mn > x) (or P(S∞ > x) and P(M∞ > x)) as x → ∞. The study of infinite number of random weighted sums was initiated by [8] in the presence of extendedly-regularly-varying-tailed Xi’s, in which they assumed, further, that θi is the product of a series of i.i.d. r.v.s {Yj, j 1}, and derived relation (2). We are interested in the asymptotic behavior for the tail probability of a random number of randomly weighted sums and their maximum defined in (1), which is motivated by two recent papers [15] and [4].

Preliminaries and main results

Proof of the main results